The Lamb-Scully paper is a good example of how even a Nobel Prize winner can occasionally write a bad paper.
The historical context is important. Einstein hypothesized the photon in 1905, but his paper was ahead of its time and was not widely accepted. For decades afterward, even once the quantum-mechanical nature of the atom was assumed by all physicists, the quantum-mechanical nature of light was considered suspect. Bohr was influential in pushing a theory in which atoms were quantized, but the light they absorbed and emitted was classical. Lamb began his career during this era.
If you read the Lamb-Scully paper, the first thing you notice is that they explicitly state that photons are absolutely necessary in order to explain phenomena such as blackbody radiation, Compton scattering, spontaneous emission, and the Lamb shift. Any internet kooks who are trying to quote Lamb and Scully as authorities against quantization of light are way off base.
As in Bohr's old-fashioned dead-end approach, they then treat the atom as a quantum-mechanical system and the electromagnetic field as a classical one. They are able to reproduce the Einstein relation $E=hf-W$, where $E$ is the maximum energy of the electron once it leaves the cathode, $h$ is the quantum-mechanical Planck's constant, $f$ is the frequency of the light, and $W$ is the energy required for the electron to escape through the surface of the cathode. This is not particularly surprising or impressive in a bastardized quantum/classical calculation like this one; essentially it just says that the light wave has to have the energy taken out of it at a resonant frequency of the atom, that frequency has to match its own frequency.
They also show that the transition rate is nonzero even when the light is first turned on, saying that their result "certainly does not imply the 'time delay' which some people used to expect for the photoelectrons produced by a classical e.m. field." This result is not as impressive as they make it sound, since the classical prediction is what one expects for a classical light wave impinging on classical atoms.
In fact, the transition rate they derive shows the real problem with their calculation. Their calculation treats every atom as independent of all the other atoms. Therefore if a classical flash of light with energy $W$ illuminates the cathode, it may ionize more than one atom, violating conservation of energy. This unphysical result shows the opposite of what they claim; it shows that their mixed quantum-classical Frankenstein fails to provide a physically acceptable explanation of the photoelectric effect. What they really need is a quantum-mechanical entanglement between the different parts of the photon's wave packet, so that if the photon is observed at atom A, it is guaranteed not to be observed at atom B. Without this quantum-mechanical "spooky action at a distance," their theory violates conservation of energy.
This issue was recognized very early on in the development of the "old" quantum theory, and it led to the Bohr-Kramers-Slater (BKS) theory, in which energy and momentum were conjectured to be conserved only on a statistical basis. Experiments as early as Bothe 1925 falsified the BKS theory by showing that when x-rays were emitted in a spherical wave into two hemispherical detectors, the two detectors were completely anticorrelated.
A modern discussion of these issues is given by Greenstein 2005. In section 2.1, they first present a summary of the Lamb-Scully argument, and then discuss the experimental verification of the existence of the anticorrelations required in order to maintain conservation of energy (Grangier 1986). The fact that this anticorrelation was not successfully observed with visible light until 1986 was due to technical limitations on the ability to produce sources of light that were eigenstates of photon number. However, the equivalent anticorrelation result with x-rays had already been demonstrated by Bothe in 1925.
One could therefore argue that the observations of the photoelectric effect were not enough to establish the existence of photons without the further verification of anticorrelations some years years later. This would be misleading, however. From the point of view of physicists reading Einstein's 1905 paper, before the quantum-mechanical nature of the atom had been established, a hybrid model such as Lamb's or the BKS theory was unavailable, and therefore the photoelectric effect really did require quantization of light. One could argue that, in the historical context of the period from 1913 (the Bohr model) to 1925 (Bothe), there was a viable BKS theory that avoided quantization of the electromagnetic field, but this is extremely misleading when modern authors such as Lamb fail to admit that nonconservation of energy was an ingredient.
Similar difficulties arise if one attempts to construct a consistent theory in which the gravitational field simply isn't quantized, unlike the other fundamental forces (Carlip 2008).
Bothe and Geiger, "Experimentelles zur Theorie von Bohr, Kramers und Slater," Die Naturwissenschaften 13 (1925) 440. The experiment is described in Bothe's 1954 Nobel Prize lecture.
Carlip, "Is quantum gravity necessary?," http://arxiv.org/abs/0803.3456
Grangier, Roger, and Aspect, "Experimental evidence for a photon anticorrelation effect on a beamsplitter," Europhys. Lett. 1 (1986) 173 -- can be found online by googling
Greenstein and Zajonc, "The quantum challenge: modern research on the foundations of quantum mechanics," Jones and Bartlett, 2005.